They have even invented a very clever name for it, namely, 'metempsychosis'.

Reincarnation

Gurdjieff taught that, contrary to conventional belief, man is not born with
a Soul, but has to create one through personal effort. He taught further that the
conventional belief that reincarnation involves the return of the soul to a
physical body on earth is wrong. He maintains that reincarnation refers to the
process of the Highest Being Body Soul entering the discarded Kesdjan Body of a
recently realized being, a process that takes place in the so called Astral
Realm. Working on developing our Highest Being Body Soul is thus not a task to leave for the future, but an essential effort to undertake in the
present, as we cannot be sure that there is an abundant supply of Kesdjan bodies
of the required gradation on the 'other side'. This process may be open to abuse
by certain 'secret societies' whose agenda could involve limiting the
development of lower caste members in order to provide a supply of Kesdjan
bodies for the benefit of a 'chosen few' Hassnamus Beings. The phenomenon of
possession could indicate the presence of an unscrupulous Hassnamus Individual
inhabiting the Kesdjan body of someone still on this side of death. (Buyer
beware!)

***

"The fact is that, in accordance with various second-order cosmic laws, the 'kesdjan being-body' cannot exist long in this sphere, and at the end of a certain time must decompose, even if the 'highest being-part' existing within it has not by that time attained the requisite degree of Reason. As long as this 'highest being-part' has not perfected its Reason to the requisite degree, it must always be dependent upon some kesdjanian arising or other, so that immediately after the second sacred rascooarno every still unperfected 'highest being-body' enters a state called 'teshgekdnel' or 'searching for a similar two-natured arising corresponding to itself,' in the hope that as soon as the highest part of another two-natured arising perfects itself to the required degree of Reason and undergoes the final process of the sacred rascooarno, and before the rapid disintegration of its kesdjan body is clearly sensed, this first 'highest being-body' might instantly enter that other kesdjan body and continue to exist in it for its further perfecting, a perfecting which sooner or later must inevitably be accomplished by every arisen 'highest being-body.'

"And that is why, in the sphere to which the higher being-parts rise after the first sacred rascooarno, that process takes place called the 'okipakhalevian substitution of the external part of the soul,' or 'substitution of a new kesdjan body for the old one.'

"Here it might as well be mentioned that your favorites also have a representation somewhat similar, as it were, to the 'okipakhalevian substitution ' They have even invented very clever names for it, 'metempsychosis' and 'reincarnation,' and in the last century have created around this question a branch of their famous 'science' which gradually became, and still is, one of those minor maleficent factors which in their totality are making their Reason, already strange enough without this, always more and more 'shooroombooroom,' as our dear Mullah Nasr Eddin would say."
(39)